Two questions about the class of matrix rings

Question

Let $(R,+,-,*,0,1)$ be an associative ring with unit $1\neq0$. Let $n$ be a positive integer greater than or equal to 2. I define a matrix ring to be a ring isomorphic to the ring of all $n$ by $n$ matrices under matrix addition, additive inverse, multiplication, additive identity, and multiplicative identity for some ring $R$ and for some positive integer $n$. Let $C$ be the class of all matrix rings. Is $C$ a first-order axiomatizable class? The second question is, whether or not $C$ is first order axiomatizable, what is an explicit axiomatization of $Th(C)$, the first-order theory of $C$.